3D positively curved space
$begingroup$
If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres.
If 2D surface is positively curved the radius of the circles at a distance $r$ from the origin becomes $R sin(frac{r}{R})$, where $R$ is the radius of the 2-Sphere.
Similarly for 3D positively curved space the radius of the spheres at a distance $r$ from the origin becomes $R sin(frac{r}{R})$,in this case whether $R$ is the radius of the 3-spheres?
curvature
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
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migrated from physics.stackexchange.com Jan 2 at 23:49
This question came from our site for active researchers, academics and students of physics.
|
show 3 more comments
$begingroup$
If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres.
If 2D surface is positively curved the radius of the circles at a distance $r$ from the origin becomes $R sin(frac{r}{R})$, where $R$ is the radius of the 2-Sphere.
Similarly for 3D positively curved space the radius of the spheres at a distance $r$ from the origin becomes $R sin(frac{r}{R})$,in this case whether $R$ is the radius of the 3-spheres?
curvature
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
$endgroup$
migrated from physics.stackexchange.com Jan 2 at 23:49
This question came from our site for active researchers, academics and students of physics.
$begingroup$
If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on.
$endgroup$
– John Rennie
Jan 2 at 8:14
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Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 9:25
$begingroup$
If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question.
$endgroup$
– John Rennie
Jan 2 at 9:46
$begingroup$
R is the radius of curvature
$endgroup$
– Reign
Jan 2 at 15:44
$begingroup$
For 3-D case is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 15:57
|
show 3 more comments
$begingroup$
If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres.
If 2D surface is positively curved the radius of the circles at a distance $r$ from the origin becomes $R sin(frac{r}{R})$, where $R$ is the radius of the 2-Sphere.
Similarly for 3D positively curved space the radius of the spheres at a distance $r$ from the origin becomes $R sin(frac{r}{R})$,in this case whether $R$ is the radius of the 3-spheres?
curvature
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
$endgroup$
If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres.
If 2D surface is positively curved the radius of the circles at a distance $r$ from the origin becomes $R sin(frac{r}{R})$, where $R$ is the radius of the 2-Sphere.
Similarly for 3D positively curved space the radius of the spheres at a distance $r$ from the origin becomes $R sin(frac{r}{R})$,in this case whether $R$ is the radius of the 3-spheres?
curvature
curvature
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
asked Jan 2 at 5:57
Apashanka DasApashanka Das
11
11
migrated from physics.stackexchange.com Jan 2 at 23:49
This question came from our site for active researchers, academics and students of physics.
migrated from physics.stackexchange.com Jan 2 at 23:49
This question came from our site for active researchers, academics and students of physics.
$begingroup$
If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on.
$endgroup$
– John Rennie
Jan 2 at 8:14
$begingroup$
Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 9:25
$begingroup$
If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question.
$endgroup$
– John Rennie
Jan 2 at 9:46
$begingroup$
R is the radius of curvature
$endgroup$
– Reign
Jan 2 at 15:44
$begingroup$
For 3-D case is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 15:57
|
show 3 more comments
$begingroup$
If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on.
$endgroup$
– John Rennie
Jan 2 at 8:14
$begingroup$
Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 9:25
$begingroup$
If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question.
$endgroup$
– John Rennie
Jan 2 at 9:46
$begingroup$
R is the radius of curvature
$endgroup$
– Reign
Jan 2 at 15:44
$begingroup$
For 3-D case is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 15:57
$begingroup$
If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on.
$endgroup$
– John Rennie
Jan 2 at 8:14
$begingroup$
If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on.
$endgroup$
– John Rennie
Jan 2 at 8:14
$begingroup$
Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 9:25
$begingroup$
Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 9:25
$begingroup$
If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question.
$endgroup$
– John Rennie
Jan 2 at 9:46
$begingroup$
If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question.
$endgroup$
– John Rennie
Jan 2 at 9:46
$begingroup$
R is the radius of curvature
$endgroup$
– Reign
Jan 2 at 15:44
$begingroup$
R is the radius of curvature
$endgroup$
– Reign
Jan 2 at 15:44
$begingroup$
For 3-D case is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 15:57
$begingroup$
For 3-D case is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 15:57
|
show 3 more comments
1 Answer
1
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oldest
votes
$begingroup$
For 3-Sphere the radius of the Curvature is R. For more information look this
Cosmology Mathematical Tripos
If you look the equations 1.1.5 and 1.1.14 that you ll see that the author defined a is the radius of the 3-sphere as "a" and used it to desribe the metric.
answered Jan 2 at 19:49
ReignReign
14918
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
For 3-Sphere the radius of the Curvature is R. For more information look this
Cosmology Mathematical Tripos
If you look the equations 1.1.5 and 1.1.14 that you ll see that the author defined a is the radius of the 3-sphere as "a" and used it to desribe the metric.
answered Jan 2 at 19:49
ReignReign
14918
$endgroup$
add a comment |
$begingroup$
For 3-Sphere the radius of the Curvature is R. For more information look this
Cosmology Mathematical Tripos
If you look the equations 1.1.5 and 1.1.14 that you ll see that the author defined a is the radius of the 3-sphere as "a" and used it to desribe the metric.
answered Jan 2 at 19:49
ReignReign
14918
$endgroup$
add a comment |
$begingroup$
For 3-Sphere the radius of the Curvature is R. For more information look this
Cosmology Mathematical Tripos
If you look the equations 1.1.5 and 1.1.14 that you ll see that the author defined a is the radius of the 3-sphere as "a" and used it to desribe the metric.
answered Jan 2 at 19:49
ReignReign
14918
$endgroup$
For 3-Sphere the radius of the Curvature is R. For more information look this
Cosmology Mathematical Tripos
If you look the equations 1.1.5 and 1.1.14 that you ll see that the author defined a is the radius of the 3-sphere as "a" and used it to desribe the metric.
answered Jan 2 at 19:49
ReignReign
14918
answered Jan 2 at 19:49
ReignReign
14918
answered Jan 2 at 19:49
ReignReign
14918
answered Jan 2 at 19:49
ReignReign
14918
14918
add a comment |
add a comment |
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$begingroup$
If you're drawing circles on a curved surface then the ratio of the radius to circumference will depend on the radius, but the exact dependence will be different for different surfaces so you need to specify what surface you are drawing the circles on.
$endgroup$
– John Rennie
Jan 2 at 8:14
$begingroup$
Ya that is on the 2-sphere, buy my question is what is R for 3-D positively curved space,is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 9:25
$begingroup$
If you're drawing 2-spheres of radius $r$ on a 3-sphere of radius $R$ then the area to radius ratio of the 2-sphere will depend on both $r$ and $R$. Off hand I don't know the exact dependence, but I think that is a maths question rather than a physics question.
$endgroup$
– John Rennie
Jan 2 at 9:46
$begingroup$
R is the radius of curvature
$endgroup$
– Reign
Jan 2 at 15:44
$begingroup$
For 3-D case is it the radius of the 3-sphere??
$endgroup$
– Apashanka Das
Jan 2 at 15:57